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Algebra and Expressions

Algebra and Expressions — Free MYP3 Mathematics Practice Questions

1QuestionCombining Like TermsConcept Practice
2 marks~3 minCriterion A
A rectangle has side lengths (2x+3)(2x + 3) cm and (x+1)(x + 1) cm.

Calculate the perimeter of the rectangle, giving your answer as a simplified expression. [2]
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2QuestionUsing the Distributive PropertyConcept Practice
2 marks~3 minCriterion B
The diagram shows three rectangles, each divided into two smaller rectangles.

Rectangle 1: width =3= 3 cm, heights xx cm and 22 cm.
Rectangle 2: width =4= 4 cm, heights xx cm and 33 cm.
Rectangle 3: width =5= 5 cm, heights xx cm and 11 cm.
a
Calculate the total area of Rectangle 1 in two different ways and show that both expressions are equal. [1]
b
State the general rule shown by this pattern in the form a(b+c)=…a(b + c) = \ldots [1]
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3QuestionCreating Equivalent Expressions through ExpansionConcept Practice
2 marks~3 minCriterion B
Each expansion below follows the same pattern.

3(x+2)=3x+63(x + 2) = 3x + 6
4(x+3)=4x+124(x + 3) = 4x + 12
5(x+4)=5x+205(x + 4) = 5x + 20
6(x+5)=6x+306(x + 5) = 6x + 30
a
Calculate the expansion of 7(x+6)7(x + 6). [1]
b
State the general rule for expanding a(b+c)a(b + c). [1]

Solutions

4QuestionCreating Equivalent Expressions through ExpansionConcept Practice
2 marks~3 minCriterion A
A rectangle has side lengths of (x+3)(x + 3) cm and (x+5)(x + 5) cm.

Calculate the simplified expression for the area of the rectangle in cm², showing all steps of your expansion. [2]
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5QuestionChecking Solutions for Word-Based ExpressionsConcept Practice
4 marks~6 minCriterion C
A student solves the word problem "a number plus three times the number equals 20" and writes:

x+3x=20x + 3x = 20

The student claims the solution is x=5x = 5.
a
Apply the substitution x=5x = 5 into the equation x+3x=20x + 3x = 20 and calculate the value of the left-hand side. [2]
b
Justify whether x=5x = 5 is a correct solution to the equation. [2]

Solutions

6QuestionCreating Algebraic Expressions from Word ProblemsConcept Practice
2 marks~3 minCriterion A
A rectangle has side lengths of (x+3)(x + 3) cm and (2x−1)(2x - 1) cm. The perimeter of a rectangle is given by

P=2(l+w)P = 2(l + w)

where ll is the length and ww is the width.
a
Write an expression for the perimeter PP in terms of xx. [1]
b
Calculate the perimeter when x=4x = 4. [1]
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7QuestionEvaluating Algebraic Formulas with Given ValuesConcept Practice
2 marks~3 minCriterion B
A square has side length ss cm and perimeter PP cm.

ss (cm)1234
PP (cm)481216
a
Write a formula for PP in terms of ss. [1]
b
Calculate the perimeter when s=5s = 5 cm and when s=10s = 10 cm. [1]

Solutions

8QuestionUsing Substitution in Real-World FormulaeConcept Practice
2 marks~3 minCriterion A
A rectangle has a length of 8 cm and a width of 5 cm.

The perimeter of a rectangle is given by

P=2(l+w)P = 2(l + w)

where ll is the length and ww is the width.

Calculate the perimeter of the rectangle. [2]
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9QuestionRecognizing Like and Unlike TermsConcept Practice
3 marks~5 minCriterion B
The scatter plot below shows the number of hours a student studies per week and their test score.
a
Describe the direction of the trend shown in the scatter plot. [1]
b
Explain what the trend tells you about the relationship between hours studied and test score. [1]
c
A student studies for 6 hours per week and scores 70. Another student studies for 10 hours per week. Using the trend in the graph, interpret what score the second student is most likely to achieve, and justify your reasoning. [1]
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10QuestionBuilding Expressions from Verbal DescriptionsConcept Practice
2 marks~3 minCriterion A
A rectangle has a length of 12 cm and a width of 5 cm.
a
Write an expression for the perimeter PP of the rectangle in terms of its length LL and width WW. [1]
b
Calculate the perimeter of the rectangle using the given dimensions. [1]
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11QuestionConstructing Expressions with BracketsConcept Practice
4 marks~6 minCriterion A
A rectangular garden has a length of (x+5)(x + 5) metres and a width of (x−2)(x - 2) metres. The owner plans to enclose the entire garden with fencing.
a
Write an expression, including brackets, for the total length of fencing required. [2]
b
Calculate the simplified form of your expression from part (a). [1]
c
The owner uses your expression to order fencing for a value of x=4x = 4. Interpret one reason why the actual fencing needed may differ from this calculated value. [1]
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12QuestionCombining Like TermsAssessment Practice
2 marks~3 minCriterion D
A student tracks weekly spending on snacks and drinks. Chips cost 2 dollars each, candy costs 1 dollar each, juice costs 3 dollars each, and water costs 1 dollar each. Let cc represent the total number of snack items (chips and candy) purchased, and dd represent the total number of drink items (juice and water) purchased. The student writes the total cost as 2c+1c+3d+1d2c + 1c + 3d + 1d.
a
Calculate the simplified expression for the total cost by combining like terms. [1]
b
Explain one limitation of using this simplified expression to model the student's actual spending. [1]
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13QuestionUsing the Distributive PropertyAssessment Practice
6 marks~9 minCriterion C
A student buys 5 identical items, each priced at xx dollars, using a coupon that gives a fixed discount of dd dollars off each item. The student writes 5(x−d)5(x - d) for the total cost.
a
Apply the distributive property to expand 5(x−d)5(x - d) and explain what each term in your result represents. [2]
b
The store instead offers a percentage discount of pp% off each item. Write an expression for the total cost of 5 items after this discount is applied. [2]
c
Compare the two models and explain whether 5(x−d)5(x - d) could still be useful when the discount is actually pp%. [2]
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14QuestionFactoring Common Terms from ExpressionsAssessment Practice
3 marks~5 minCriterion C
A rectangle has a fixed length of 3 units. Its area AA (in square units) is given by A=3wA = 3w, where ww is the width in units.

The graph shows AA against ww for 0≤w≤100 \leq w \leq 10.
a
Describe what the graph of A=3wA = 3w looks like. [1]
b
Explain why AA is directly proportional to ww. [1]
c
A student claims the constant of proportionality is 10 because the graph reaches A=30A = 30 when w=10w = 10. Calculate the correct constant of proportionality and explain the student's error. [1]
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15QuestionCreating Equivalent Expressions through ExpansionAssessment Practice
6 marks~9 minCriterion D
A rectangular garden bed has a length of (x+4)(x + 4) m and a width of 33 m. A student expands the expression for the area to obtain 3x+123x + 12.
a
Calculate the area of the garden bed for x=2x = 2 using both 3(x+4)3(x + 4) and 3x+123x + 12. [2]
b
The actual width is 3.13.1 m due to a measurement error. Calculate the actual area for x=2x = 2. [1]
c
The student claims 3x+123x + 12 is still a valid model for the garden bed. Analyse this claim using the percentage difference in area. [3]
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16QuestionCreating Algebraic Expressions from Word ProblemsAssessment Practice
5 marks~8 minCriterion B
A tiling pattern is made of square tiles arranged in rows. The first row has 1 tile, the second row has 3 tiles, the third row has 5 tiles, and the fourth row has 7 tiles.
a
Describe the pattern in the sequence 1,3,5,7,…1, 3, 5, 7, \ldots and write an algebraic expression for the number of tiles in row nn. [2]
b
Calculate the number of tiles in the 20th row. [1]
c
A designer claims that row 50 will have more than 100 tiles. Justify whether this claim is correct. [2]

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17QuestionChecking Solutions for Word-Based ExpressionsAssessment Practice
4 marks~6 minCriterion D
A student models the total cost of a school bake sale using the equation C=2.50n+15C = 2.50n + 15, where nn is the number of cupcakes sold and 1515 dollars covers fixed costs such as table rental.
a
Calculate the total cost when n=10n = 10. [1]
b
Explain one assumption made by this model. [1]
c
A classmate says the model must be wrong because 40 dollars for 10 cupcakes is too expensive. Analyse whether this objection identifies a mathematical error or a real-world limitation, and justify your conclusion. [2]
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18QuestionSubstituting Values into Simple ExpressionsAssessment Practice
2 marks~3 minCriterion C
A fruit seller charges 3 dollars per kilogram of apples plus a fixed delivery fee of 2 dollars. Let cc represent the total cost in dollars and kk represent the number of kilograms of apples.
a
Write an expression for cc in terms of kk. [1]
b
Describe what the term 3k3k and the constant 22 each represent in your expression. [1]

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19QuestionEvaluating Algebraic Formulas with Given ValuesAssessment Practice
8 marks~12 minCriterion D
A civil engineer uses the formula V=lwhV = lwh to calculate the volume of concrete needed for a rectangular foundation slab, where ll is length, ww is width, and hh is height. The design dimensions are l=12.0l = 12.0 m, w=8.0w = 8.0 m, and h=0.50h = 0.50 m. Each dimension has a tolerance of ±5%\pm 5\%. Concrete costs 120 dollars per cubic metre.
a
Calculate the design volume of concrete needed. [2]
b
Calculate the minimum and maximum possible volumes of concrete, taking into account the ±5%\pm 5\% tolerance on each dimension. Show your working. [3]
c
Analyse how the volume uncertainty affects the cost of the concrete and explain one limitation of modelling the foundation slab as a rectangular prism. [3]
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20QuestionBuilding Expressions from Verbal DescriptionsAssessment Practice
5 marks~8 minCriterion D
A student running a school bake sale writes the expression 5n+205n + 20 to model total profit in dollars, where nn is the number of cakes sold at 5 dollars each and 20 dollars is the cost of ingredients.
a
Describe two assumptions the student made when building this model. [2]
b
Explain how unsold cakes and changes in ingredient costs would each affect the accuracy of this model. [2]
c
Analyse one additional real-world factor not captured by 5n+205n + 20 and explain how including it would change the expression. [1]
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21QuestionRecognizing Like and Unlike TermsAssessment Practice
6 marks~9 minCriterion C
A student writes the expression 3a+2b+5a3a + 2b + 5a to model the total cost, in dollars, of buying apples and bananas for a class party. Here aa is the cost per kilogram of apples and bb is the cost per kilogram of bananas. The student claims that 3a3a and 5a5a are unlike terms because they have different coefficients, so the expression cannot be simplified.
a
Identify the error in the student's reasoning and explain why 3a3a and 5a5a are like terms. [2]
b
Calculate the simplified expression that correctly models the total cost. [2]
c
The model assumes that apples always cost aa dollars per kilogram regardless of the quantity bought. Analyse one way this assumption could make the model inaccurate in a real shopping situation. [2]
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22QuestionCreating Expressions from Situational ProblemsAssessment Practice
5 marks~8 minCriterion C
A school drama club sells tickets for a show at 8 dollars each and spends 50 dollars on advertising. A student writes the profit expression P=8x−50P = 8x - 50, where xx is the number of tickets sold.
a
Write the value of PP when no tickets are sold. Explain what this value means in context. [2]
b
Calculate the profit when 120 tickets are sold. [2]
c
The student claims that selling 80 tickets will produce twice the profit of selling 40 tickets. Analyse whether this claim is correct. [1]

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23QuestionConstructing Expressions with BracketsAssessment Practice
5 marks~8 minCriterion B
When a bracket of the form (x+n)2(x + n)^2 is expanded, a pattern emerges in the terms produced.
a
Expand each expression and complete the table below. [2]

Expression: (x+1)2(x+1)^2, (x+2)2(x+2)^2, (x+3)2(x+3)^2, (x+4)2(x+4)^2

Constant term: ___ , ___ , ___ , ___

Coefficient of xx: ___ , ___ , ___ , ___
b
Describe the relationship between nn and each of the two values you recorded in part (a). [1]
c
A student claims that (x+7)2=x2+14x+49(x + 7)^2 = x^2 + 14x + 49. Using your answer from part (b), explain whether the student is correct. [2]

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24QuestionCreating Expressions from Situational ProblemsAssessment Practice
5 marks~8 minCriterion D
A student is designing a rectangular garden. The length is 3 m more than twice the width. To stay within the fencing budget, the perimeter must not exceed 30 m. The student writes the inequality 2(w+(2w+3))≤302(w + (2w + 3)) \leq 30.
a
Explain what the expression 2(w+(2w+3))2(w + (2w + 3)) represents in this context. [1]
b
Calculate the perimeter when w=4w = 4 m, and determine whether this width satisfies the budget constraint. [2]
c
Analyse one limitation of using this algebraic model for real-world garden planning. [2]
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